\section{1.5. Relations with PDE} 
\begin{frame}[allowframebreaks]{1.5. }

\vspace{-0.4cm}

1.5. Relations with PDE. 

(1) Let $X$ be affine. 

(2) For simplicity, assume $\Theta_X$ is spanned by $n = \dim X$ commuting vector fields $\partial_i$. 

(3) Consider a differential system of $p$ equations in $q$ unknowns on $X$
\begin{equation}
\sum_{j=1}^q P_{ij} u_j = 0 \quad (1 \le i \le p)
\label{eq-5-1}
\end{equation}
where $P_{ij} \in \mathcal{D}(X)$, i.e. $P_{ij}$ is a polynomial in the $\partial^\alpha$, with coefficients in $\mathcal{O}(X)$. 

(4) To (\ref{eq-5-1}) we associate a morphism of $\mathcal{D}(X)$-modules
\[
\pi: \mathcal{D}(X)^p \to \mathcal{D}(X)^q
\]
defined by
\begin{equation}
(R_1, \ldots, R_p) \mapsto (\Sigma_i R_i P_{i1}, \ldots, \Sigma_i R_i P_{iq}). 
\end{equation}

(5) The image $J$ of $\pi$ is a left submodule. 

(6) To (\ref{eq-5-1}) we associate the $\mathcal{D}(X)$-module $M = \mathcal{D}(X)^q / J$. 

(7) We have then, by definition, an exact sequence
\begin{equation}
\mathcal{D}(X)^p \to \mathcal{D}(X)^q \to M \to 0.
\label{eq-5-3}
\end{equation}

(8) In particular, $M$ is coherent. 

(9) Conversely, let $M$ be a coherent $\mathcal{D}(X)$-module. 

(10) There is then an exact sequence (\ref{eq-5-3}). 

(11) If $(u_i)$ is the canonical basis of $\mathcal{D}(X)^q$, then $J$ is generated by $p$ elements which can be written as the left-hand sides in (\ref{eq-5-1}) and then (\ref{eq-5-1}) is associated to $M$. 

(12) As a consequence, there is a natural correspondence between coherent $\mathcal{D}(X)$-modules and systems of linear PDE.

(13) To a $q$-tuple $(f_i)$ of regular functions is associated a $\mathcal{D}(X)$-morphism $\mathcal{D}(X)^q \to \mathcal{O}(X)$ which maps $u_i$ to $f_i$ ($1 \le i \le q$). 

(14) This morphism can be factored through $M$ if and only if the $f_i$ are solutions of the system (\ref{eq-5-1}). 



(15) The space of solutions in $\mathcal{O}(X)$ of (\ref{eq-5-1}) can be therefore identified with $$\mathrm{Sol}\, M = \mathrm{Hom}_{\mathcal{D}(X)}(M, \mathcal{O}(X)). $$

(16) More generally, the space of solutions of (\ref{eq-5-1}) in any $\mathcal{D}(X)$-module $F$ can be identified with $$\mathrm{Hom}_{\mathcal{D}(X)}(M, F). $$

(17) These considerations go over to sheaves, and show that $\mathrm{Hom}_{\mathcal{D}_X}(M, \mathcal{O}_X)$ is the sheaf of germs of solutions of the system of PDE which represents $M$ locally as above. 

(18) $\mathrm{Sol}\, M$ is a sheaf of $\mathbb{C}_X$-modules, still with respect to the Zariski-topology. 

(19) It is however more usual, and more useful, to pass to the analytic topology and consider the sheaf of germs of solutions in the latter. 


(20) This passage, already discussed in IV (B. Malgrange. Regular connections - after Deligne), will be taken up again later when we set up the Riemann-Hilbert correspondence. 

(21) More generally, one will consider the complex $\mathrm{Sol}^{\bullet} M = \mathrm{RHom}_{\mathcal{D}_X}(M, \mathcal{O}_X)$ of ``generalized solutions'' of $M$, again going over to the ordinary topology. 

(22) We do not pursue that here, but mention it since it gives one of the motivations for passing from the system (\ref{eq-5-1}) to a $\mathcal{D}$-module.


\newpage 

{\color{red}Notebook.}

1. After studying this section, what questions or problems can I ask, and work with?





\newpage 

This section (1.5) beautifully explains the dictionary between systems of linear partial differential equations (PDEs) and coherent \(\mathcal{D}\)-modules. 

It sets the stage for the modern, algebraic approach to differential equations—central to the Riemann-Hilbert correspondence and D-module theory.

Below is a curated list of questions, exercises, and research-style problems you can explore to deepen your understanding. 

They range from computational to conceptual to theoretical, grouped by theme.


\newpage 

I. Understanding the Dictionary: From PDEs to \(\mathcal{D}\)-modules

(1) Explicit computation:  

Take a concrete PDE system on \(X = \mathbb{A}^1\) or \(\mathbb{A}^2\), e.g.  
\[
\frac{d}{dx} u = 0 \quad \text{or} \quad \left(\frac{\partial}{\partial x} - \frac{\partial}{\partial y}\right) u = 0.
\]  

Write down the corresponding matrix \((P_{ij})\), construct the map \(\pi: \mathcal{D}^p \to \mathcal{D}^q\), and compute \(M = \mathcal{D}^q / \mathrm{im}(\pi)\).  

$\to$ What is \(\mathrm{Hom}_{\mathcal{D}}(M, \mathcal{O})\)? Verify it matches the solution space.

(2) Nontrivial example:  

Consider the Cauchy–Euler equation on \(\mathbb{A}^1 \setminus \{0\}\):  
\[
x \frac{d}{dx} u = \lambda u, \quad \lambda \in \mathbb{C}.
\]  

How do you encode this as a \(\mathcal{D}\)-module on \(\mathbb{A}^1\)? (Hint: use the module \(\mathcal{D} / \mathcal{D} \cdot (x \partial_x - \lambda)\).)

(3) System vs single equation:  

Show that a system of PDEs can always be reduced to a single higher-order PDE (in one unknown) only in special cases. Why does the \(\mathcal{D}\)-module framework avoid this limitation?


\newpage 

II. From \(\mathcal{D}\)-modules back to PDEs

(4) Reconstruction:  

Let \(M = \mathcal{D} / \mathcal{D} \cdot P\) for some \(P \in \mathcal{D}(X)\). Write down the corresponding PDE.  

What if \(M = \mathcal{D}^2 / \langle (P_1, P_2) \rangle\)? How many unknowns? How many equations?

(5) Minimal presentation:  

Given a coherent \(\mathcal{D}\)-module \(M\), the presentation \(\mathcal{D}^p \to \mathcal{D}^q \to M \to 0\) is not unique.  

$\to$ What does changing the presentation correspond to in PDE terms? (Answer: adding redundant equations or unknowns, gauge transformations.)

\newpage 

III. Solution Functors and Representability

(6) Why \(\mathrm{Hom}_{\mathcal{D}}(M, \mathcal{O})\)?  

Prove that a \(\mathcal{D}\)-linear map \(\phi: M \to \mathcal{O}(X)\) is uniquely determined by the images of the generators \(u_1, \dots, u_q\), and that these images must satisfy the PDE system.  

$\to$ This justifies (15).

(7) Functoriality:  

If \(M \to N\) is a morphism of \(\mathcal{D}\)-modules, how does it induce a map on solution spaces?  

$\to$ Interpret this as “restriction of solutions” or “change of variables”.

(8) Non-\(\mathcal{O}\)-valued solutions:  

Let \(F = \mathcal{C}^\infty_X\) (smooth functions) or \(\mathcal{O}_X^{\mathrm{an}}\) (analytic functions). Why is \(\mathrm{Hom}_{\mathcal{D}}(M, F)\) still the solution space?  

$\to$ Note: \(\mathcal{D}\) acts naturally on these.


IV. Sheafification and Topology

(9) Zariski vs analytic topology:  

Why is the sheaf \(\mathrm{Hom}_{\mathcal{D}_X}(M, \mathcal{O}_X)\) in the Zariski topology often too small?  

$\to$ Example: On \(\mathbb{A}^1\), the equation \(\partial u = u\) has no regular (polynomial) solutions, but has analytic solution \(e^x\).  

$\to$ So \(\mathrm{Hom}_{\mathcal{D}}(M, \mathcal{O}) = 0\), but analytic solutions exist.

(10) Sheaf of solutions:  

Show that \(\mathcal{S}ol(M) := \mathrm{Hom}_{\mathcal{D}_X}(M, \mathcal{O}_X^{\mathrm{an}})\) is a local system (locally constant sheaf) when \(M\) is ``regular holonomic'' (preview of Riemann-Hilbert).


\newpage 

V. Conceptual and Theoretical Questions

(11) Why coherent?  

Why do we restrict to *coherent* \(\mathcal{D}\)-modules? What goes wrong if \(M\) is not coherent?  

$\to$ Answer: Non-coherent modules correspond to “infinite systems” or “ill-posed” PDEs; solution spaces may not be finite-dimensional or well-behaved.

(12) What is lost in algebraization?  

The passage from analytic PDEs to algebraic \(\mathcal{D}\)-modules forgets analytic information (e.g., growth conditions). How is this recovered later? 

$\to$ Answer: via `regularity' and `Riemann-Hilbert'.

(13) Duality:  

The solution functor is \(\mathrm{Hom}_{\mathcal{D}}(-, \mathcal{O})\). Why is this left exact but not right exact?  

$\to$ This motivates derived solutions: \(\mathrm{RHom}_{\mathcal{D}}(M, \mathcal{O})\) in (21), which captures *generalized solutions* (e.g., distributions, hyperfunctions).


\newpage 

VI. Advanced Explorations (for later)

(14) Holonomic modules:  

A coherent \(\mathcal{D}\)-module \(M\) is *holonomic* if its characteristic variety has minimal dimension.  

$\to$ Show that for holonomic \(M\), the solution space (analytic) is finite-dimensional over \(\mathbb{C}\).  

$\to$ This is a cornerstone of the theory.

(15) Riemann–Hilbert correspondence:  

When \(M\) is regular holonomic, \(\mathcal{S}ol(M)\) is a perverse sheaf (in fact, a local system if \(M\) is smooth).  

$\to$ The Riemann–Hilbert functor \(M \mapsto \mathcal{S}ol(M)\) is an equivalence of categories.  

$\to$ How does your PDE example fit into this?

(16) Computational D-module theory:  
    
Use software (e.g., Macaulay2 with `Dmodules' package, or Singular) to compute:
    
$\to$ Annihilator of a function (e.g., \(e^{x^2}\)),
    
$\to$ Solution space of a given \(\mathcal{D}\)-module.

\newpage 

VII. Philosophical / Motivational

(17) Why replace PDEs by modules?  

$\to$ Modules allow functorial operations: pullback, pushforward, duality.

$\to$ They unify ODEs, PDEs, integrable systems.

$\to$ They enable homological methods (Ext, Tor, derived categories).

$\to$ They connect analysis (PDEs) to topology (sheaves) via Riemann–Hilbert.


(18) Historical note:  

This algebraic approach was pioneered by Kashiwara, Bernstein, Malgrange, and Deligne in the 1970s-80s.  
    
How does this compare to classical PDE theory (e.g., Cauchy-Kovalevskaya)?


\newpage 

Suggested Mini-Project

Take the wave equation \(\partial_t^2 u - \partial_x^2 u = 0\) on \(\mathbb{A}^2\).  

Encode it as a \(\mathcal{D}\)-module \(M\).  

Compute \(\mathrm{Hom}_{\mathcal{D}}(M, \mathcal{O})\) (algebraic solutions).  

Compare with analytic solutions \(f(x+t) + g(x-t)\).  

Discuss why algebraic solutions are only polynomials in \(x \pm t\).


By working through these questions, you'll move from passive understanding to active mastery of the bridge between differential equations and algebraic geometry — the heart of D-module theory.


\end{frame}

